There is never a case where a TW should be treated as a Woman, except there is, I work in a large organisation, there are several TW, TM and NB people, on a day to day basis in normal working relationships it does no harm to anyone to refer to them by the name and pronouns they have chosen. Does that mean I think that TW belong in female prisons, changing rooms, hospital wards etc, no.
I think we need to stop doing this, not to reject trans people's sense of self, but because we need to go back to the start and start again with different words.
I'm happy to believe trans people (and indeed those who actively identify as cis) feel a genuine thing. I can understand how they may see something in the opposite sex, or in society's construction of the opposite sex, and feel they understand themselves better through that lens. But these are not the same things as being the opposite sex. So the two things.. the feeling and the sex, should not have the same name.
If the two things, the feeling and the sex, had different names, different pronouns, there would never have been any question about whether single sex provisions should be opened up based on gender. There's no reason behind it, no coherent argument. The demand exists only because the same name was used for two different concepts.
I can see how at the start, when it was transsexual (medically transitioning) people saying "I should have been the opposite sex", it seemed like the right thing to do. But the more we understand about trans people and the different ways they think about and embody their gender, the clearer it is that this is not some sort of mental shadowing of body sex, but a different thing altogether. IMO linking this feeling to sex has been a red herring. It's actually limiting our understanding.
"Gender" identity is, I think, more about finding a metaphor to express your inner sense of who you are and to ask people to understand that about you.
The cat genders and so on get short shrift but I think they may be the way forward. Not because anyone is really a cat, but because non-human neo gender identities could offer a social language of archetypes and metaphor to express self without mapping yourself onto other humans' real lives.